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SHEAR IN BEAMS

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Example (4.1): A rectangular beam has the dimensions shown in Figure 4.12.a and is loaded with a 40 ton concentrated factored load at the middle of the span. Design the necessary shear reinforcement given that Width of support is equal to 30 cm. Solution : 1- Draw shearing force diagram: Shearing force diagram is shown in Figure 4.12.b 2- Calculate the shear capacity of concrete:

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**(a) Beam and loads; (b) shear force diagram; (c) bending moment diagram**

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3- Check whether the dimensions of the cross section are adequate for ensuring a ductile mode of failure:

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**4- Classify the factored shear force:**

5- Check stirrups spacing:

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**6- The shear reinforcement is drawn in Figure 4.12.d.**

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Shear reinforcement

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Example (4.2): A rectangular beam has the dimensions shown in Figure 4.13.a and is loaded with a uniform service dead load of 6.20 t/m (including own weight of beam). Design the necessary shear reinforcement given that Width of support is equal to 30 cm. (a) Beam and loads; (b) shearing force diagram

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**Solution : 1- Draw shearing force diagram:**

The shearing force diagram is shown in Figure 4.13.b Shear force at the centerline of the support is given as: Critical section for shear is located at a distance of 54 cm from the face of support. From similar triangles, design shear force is equal to tons. 2- Calculate the shear capacity of concrete:

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**4- Classified the factored shear forces:**

3- Check whether the dimensions of the cross section are adequate for ensuring a ductile mode of failure: 4- Classified the factored shear forces: The forces are shown in Figure 4.13.c. Shear force zone classification

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**Maximum stirrup spacing is not to exceed the smaller of d/2 = 27**

Maximum stirrup spacing is not to exceed the smaller of d/2 = 27.0 cm or 60 cm. Use φ 8 mm vertical stirrups spaced at 14 cm.

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**5- Check stirrups spacing**

Stirrups spacing is checked for all the zones. Shear reinforcement 6- The shear reinforcement is shown in Figure 4.13.d

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Example (4.3): A reinforced concrete beam 35 cm × 50 cm in cross section , shown in Figure 4.14.a, is supported over a span of 6 m. It carries a concentrated factored load of 15 tons at a distance of 0.50 m from one end in addition to a factored live load of 1.80 t/m. Calculate the shear reinforcement if a) only vertical stirrups are used, b) two bent bars at 45 degree at the same cross section, and c) two groups of bars are bent at 45 degree, each group consisting of two bars. The two groups are separated by a distance of 45 cm. Width of support is equal to 25 cm. Solution :

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**(a) Beam and loads; (b) shearing force diagram**

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**A. When only vertical stirrups are used**

i.e. cross sectional dimensions are adequate in terms of preventing brittle mode of failure.

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**Maximum stirrup spacing:**

Maximum stirrup spacing is not to exceed the smaller of d/2 = cm and 60 cm. Use φ 8 mm vertical stirrups spaced at 15 cm.

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**B. When two bars are bent at the same cross section**

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**C. When two groups are bent at 45 degree separated by 45 cm**

(continued); (c) Two bars are bent-up; (d) four bars are bent-up

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